Block Disentanglement in CRL: Bridging Identifiability and Visual State Estimation

Emre Acartürk; Pranamya Kulkarni; Puranjay Datta; Karthikeyan Shanmugam; Burak Varıcı; Ali Tajer

cs.LGOct 5, 2026

Block Disentanglement in CRL: Bridging Identifiability and Visual State Estimation

Authors: Emre Acartürk, Pranamya Kulkarni, Puranjay Datta, Karthikeyan Shanmugam, Burak Varıcı, Ali Tajer

Organizations: Rensselaer Polytechnic Institute Troy, NY, USA · Swiss AI, EPFL Lausanne, Switzerland · University of Delaware Newark, DE, USA

Abstract

Causal representation learning (CRL) is the process of recovering causally-related latent variables from high-dimensional observations. As a label-free inference method, CRL is particularly attractive for applications where data labels are unavailable or impractical to obtain. While there has been significant progress in understanding the identifiability guarantees of CRL, such guarantees often hold under highly stylized assumptions, which temper the direct application to real-world problems. This paper has a two-fold objective for interventional CRL. First, it establishes identifiability guarantees for substantially weaker interventional assumptions, resulting in block disentanglement of the causal variables, where the block structure depends on the realistically available intervention mechanisms. Secondly, the block disentanglement framework is used for embodied visual state estimation, in which the objective is to recover the latent physical variables of a robotic system directly from visual data (images and videos) without labeled data. These two components are critically complementary. The block disentanglement theory delineates identifiability guarantees under weakened assumptions, and the application demonstrates that the resulting objective remains effective in a controlled embodied setting despite further assumption violations, providing a theory-to-practice bridge needed to translate the promise of label-free CRL into practical problems.

Figures & tables

Figure 1 : Training pipeline. A shared Autoencoder 1 (AE1) compresses each $128\times 128\times 1$ image $X$ into an $8\times 8\times 1$ feature vector $Y$ . Features from both cameras are concatenated along the channel axis, forming an $8\times 8\times 2$ input to the LDR and AE2. AE2 then outputs the final $d_{b}\times 1$ latent estimate $\smash{\hat{Z}}$ .

Model	Training Labels	Joint 1	Joint 2	Joint 3	Joint 4	Joint 5	Joint 6
Ours	none during training	0.083	$0.015$	$0.217$	$0.035$	$0.198$	$0.080$
RoboPEPP	$1\%$ of dataset	0.136	$0.039$	$0.237$	$0.053$	$0.253$	$0.200$
	$5\%$ of dataset	0.075	$0.017$	$0.134$	$0.024$	$0.153$	$0.081$
	$10\%$ of dataset	0.030	$0.010$	$0.072$	$0.022$	$0.091$	$0.063$
	$100\%$ of dataset	0.003	$0.001$	$0.007$	$0.003$	$0.010$	$0.011$

Table 1 : Comparison of MSE (

\text{rad}^{2}

) for RoboPEPP and our method in assumption-matched regime.

$\lambda$	$d_{b}$	Active $R^{2}(Z_{\mathcal{I}}\mid\smash{\hat{Z}_{\hat{\mathcal{I}}}})$	All-latent $R^{2}(Z_{\mathcal{I}}\mid\smash{\hat{Z}})$	Feature $R^{2}(Z_{\mathcal{I}}\mid Y)$
0	16	0.33	0.72	0.77 (AE1 frozen)
50	16	0.58	0.63
100	16	0.63	0.66
0	32	0.23	0.77
50	32	0.61	0.67
100	32	0.55	0.66

Table 2 : Embodied setting results across score regularization weight

\lambda

and bottleneck dimension

d_{b}

Appendix figures & tables33 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 2 : Training dynamics across bottleneck dimensions $d_{b}$ ( $\lambda=50$ ).

Figure 3 : Training dynamics across score-regularization weights $\lambda$ ( $d_{b}=8$ ).

Figure 4 : Training dynamics across score-regularization weights $\lambda$ ( $d_{b}=16$ ).

Figure 5 : Training dynamics across score-regularization weights $\lambda$ ( $d_{b}=32$ ).

Figure 6 : $R^{2}$ under multi-dominant-joint data ( $d_{b}=16$ )

Figure 7 : The six primary joint angles of the Franka Emika Panda arm targeted for intervention, together with their respective axes of rotation. Note that the final (7th) joint is kept fixed.

Joint	Scenario	Distribution
2	Observational	$\mathcal{TN}_{[-1.5,\,1.5]}(0,\,1)$
	Intervention 1	$\mathcal{TN}_{[-1.5,\,1.5]}(-0.75,\,0.5)$
	Intervention 2	$\mathcal{TN}_{[-1.5,\,1.5]}(\;\;0.75,\,0.5)$
4	Observational	$\mathcal{TN}_{[-1.5,\,1.5]}(0,\,1)$
	Intervention 1	$\mathcal{TN}_{[-1.5,\,1.5]}(-0.75,\,0.5)$
	Intervention 2	$\mathcal{TN}_{[-1.5,\,1.5]}(\;\;0.75,\,0.5)$

Appendix

Table 3: Sampling distributions for observational and interventional settings for single camera setup.

Joint	Scenario	Distribution
1	Observational	$\mathcal{TN}_{[0,\,3]}(1.2,\,0.4)$
	Intervention 1	$\mathcal{TN}_{[0,\,3]}(2.0,\,0.4)$
	Intervention 2	$\mathcal{TN}_{[0,\,3]}(\;\;0.6,\,0.4)$
2	Observational	$\mathcal{TN}_{[-1.5,\,1.5]}(0,\,0.4)$
	Intervention 1	$\mathcal{TN}_{[-1.5,\,1.5]}(0.7,\,0.4)$
	Intervention 2	$\mathcal{TN}_{[-1.5,\,1.5]}(\;\;-0.7,\,0.4)$

Appendix

Table 4: Sampling distributions for the in-distribution (ID) dataset. These truncated normal distributions define the observational and interventional data used for the two-camera Independent and Occlusion experiments.

Joint	OOD Observational Distribution
1	$\mathcal{TN}_{[0,\,3]}(1.2,\,0.4)$
2	$\mathcal{TN}_{[-1.5,\,1.5]}(0.0,\,0.4)$
3	$\mathcal{TN}_{[-1.5,\,1.5]}(0.0,\,0.4)$
4	$\mathcal{TN}_{[-1.5,\,1.5]}(0.8,\,0.4)$
5	$\mathcal{TN}_{[-1.5,\,1.5]}(0.0,\,0.4)$
6	$\mathcal{TN}_{[0,\,3]}(0.5,\,0.4)$

Appendix

Table 5 : Observational sampling distributions for the out-of-distribution (OOD) dataset. These parameters define the OOD test sets for the Two-Camera Independent, Causal, and Occlusion experiments. The interventional distributions for the OOD dataset remain identical to those of the in-distribution dataset, as defined in Table 4 .

Figure 8 : The causal model of the robot joints used to generate the dataset. In this graph, J i represents the angle of joint i .

Joint	Scenario	Distribution
1	Observational	$\mathcal{TN}_{[0,\,3]}(1.2,\,0.4)$
	Intervention 1	$\mathcal{TN}_{[0,\,3]}(2.0,\,0.4)$
	Intervention 2	$\mathcal{TN}_{[0,\,3]}(0.6,\,0.4)$
2	Observational	$\mathcal{TN}_{[-1.5,\,1.5]}(0,\,0.4)$
	Intervention 1	$\mathcal{TN}_{[-1.5,\,1.5]}(0.7,\,0.4)$
	Intervention 2	$\mathcal{TN}_{[-1.5,\,1.5]}(-0.7,\,0.4)$

Appendix

Table 6: Sampling distributions for observational and interventional settings for causal dataset corresponding to the causal graph Figure 8

	Joint 1		Joint 2		Joint 3		Joint 4		Joint 5		Joint 6
Model Setup	MCC	MSE	MCC	MSE	MCC	MSE	MCC	MSE	MCC	MSE	MCC	MSE
1C, indep.	–	–	$0.949$	$0.053$	–	–	$0.975$	$0.029$	–	–	$0.957$	$0.049$
2C, indep.	$0.874$	$0.083$	$0.979$	$0.015$	$0.634$	$0.217$	$0.950$	$0.035$	$0.679$	$0.198$	$0.884$	$0.080$
2C, causal	$0.921$	$0.058$	$0.966$	$0.020$	$0.788$	$0.106$	$0.976$	$0.019$	$0.742$	$0.051$	$0.756$	$0.070$
2C, indep., occl.	$0.844$	$0.101$	$0.964$	$0.025$	$0.568$	$0.245$	$0.884$	$0.082$	$0.617$	$0.225$	$0.768$	$0.145$

Appendix

Table 7 : MCC and MSE in assumption-matched regime. MSE is reported in radians squared.

Panel of Figure 9 — Figure 9 : Single Camera: Scatter-plots of ground-truth vs. estimated angles for joints 2, 4, and 6

Panel of Figure 10 — Figure 10 : Scatter plots evaluating the two-camera Independent model on the in-distribution (ID) test set ( Table 4 ) . Each plot visualizes the relationship between a learned latent variable and its corresponding ground-truth joint angle. The displayed Correlation and MSE values correspond to the single best trial out of 15 runs, while the results presented in Tables 7 and 9 correspond to the mean statistics.

Panel of Figure 11 — Figure 11 : Scatter plots evaluating the two-camera causal model on the causally-generated test set. Each plot visualizes the relationship between a learned latent variable and its corresponding ground-truth joint angle. The displayed Correlation and MSE values correspond to the single best trial out of 15 runs, while the results presented in Tables 7 and 9 correspond to the mean statistics.

Experiment	Distribution	Patch Size	Joint 1	Joint 2	Joint 3	Joint 4	Joint 5	Joint 6
Independent	ID	–	0.083	0.015	0.217	0.035	0.198	0.080
Causal	ID	–	0.058	0.020	0.106	0.019	0.051	0.070
Occlusion	ID	16	0.089	0.018	0.225	0.044	0.200	0.082
Occlusion	ID	32	0.101	0.025	0.245	0.082	0.225	0.145
Occlusion	ID	64	0.186	0.077	0.322	0.258	0.298	0.273
Independent	OOD	–	0.084	0.017	0.239	0.048	0.199	0.120

Appendix

Table 8 : Mean Squared Error (MSE) in radians squared (

\text{rad}^{2}

) for each joint under various experimental conditions for two camera angles. The table compares performance on in-distribution (ID) test set ( Table 4 ) and out-of-distribution (OOD) test set ( Table 5 ) for independent, causal, and occluded inference models.

	2C, indep.		2C, causal
Joint Angle	MCC	MSE	MCC	MSE
Joint 1	$0.874\pm 0.004$	$0.083\pm 0.006$	$0.921\pm 0.003$	$0.058\pm 0.003$
Joint 2	$0.979\pm 0.001$	$0.015\pm 0.001$	$0.966\pm 0.002$	$0.020\pm 0.001$
Joint 3	$0.634\pm 0.010$	$0.217\pm 0.011$	$0.788\pm 0.003$	$0.106\pm 0.005$
Joint 4	$0.950\pm 0.002$	$0.035\pm 0.002$	$0.976\pm 0.001$	$0.019\pm 0.001$
Joint 5	$0.679\pm 0.010$	$0.198\pm 0.011$	$0.742\pm 0.006$	$0.051\pm 0.003$

Appendix

Table 9 : Comparison of MCC and MSE (

\text{rad}^{2}

) for each joint across two model settings with error bars. Mean and Std Dev are calculated across the 15 runs. Values are reported as Mean ± Std Dev.

Dataset	Distribution	Epochs	Patch Size	Joint 1	Joint 2	Joint 3	Joint 4	Joint 5	Joint 6
2.6k	ID	100	-	0.136	0.039	0.237	0.053	0.253	0.200
2.6k	ID	100	16	0.166	0.030	0.287	0.072	0.309	0.218
2.6k	ID	100	32	0.186	0.060	0.305	0.125	0.331	0.263
2.6k	ID	100	64	0.412	0.358	0.462	1.328	0.515	0.573
2.6k	OOD	100	-	0.202	0.054	0.374	0.101	0.378	0.454
2.6k	OOD	100	16	0.220	0.058	0.413	0.117	0.412	0.432

Appendix

Table 10 : Per-joint Mean Squared Error (MSE) for the RoboPEPP model in radians squared (

\text{rad}^{2}

). The table presents an ablation study on the effect of varying training data labels, evaluated on both in-distribution (ID) test set ( Table 4 ) and out-of-distribution (OOD) test set ( Table 5 ).

Model	Experiment	Patch Size	Joint 1	Joint 2	Joint 3	Joint 4	Joint 5	Joint 6
Our method	Independent	–	$0.083$	$0.015$	$0.217$	$0.035$	$0.198$	$0.080$
	Causal	–	$0.058$	$0.020$	$0.106$	$0.019$	$0.051$	$0.070$
	Occlusion	16	$0.089$	$0.018$	$0.225$	$0.044$	$0.200$	$0.082$
	Occlusion	32	$0.101$	$0.025$	$0.245$	$0.082$	$0.225$	$0.145$
	Occlusion	64	$0.186$	$0.077$	$0.322$	$0.258$	$0.298$	$0.273$
RoboPEPP	2.6k Dataset	–	$0.136$	$0.039$	$0.237$	$0.053$	$0.253$	$0.200$

Appendix

Table 11 : Comparison of In-Distribution (ID) Mean Squared Error (MSE) in radians squared (

\text{rad}^{2}

) for the our score-based method and RoboPEPP models. Results are shown per joint under various experimental conditions.

Model	Experiment	Patch Size	Joint 1	Joint 2	Joint 3	Joint 4	Joint 5	Joint 6
Our method	Independent	–	$0.084$	$0.017$	$0.239$	$0.048$	$0.199$	$0.120$
	Causal	–	$0.108$	$0.044$	$0.241$	$0.085$	$0.219$	$0.116$
	Occlusion	16	$0.092$	$0.019$	$0.249$	$0.054$	$0.205$	$0.109$
	Occlusion	32	$0.103$	$0.024$	$0.288$	$0.079$	$0.226$	$0.140$
	Occlusion	64	$0.201$	$0.087$	$0.356$	$0.227$	$0.331$	$0.216$
RoboPEPP	2.6k Dataset	–	$0.202$	$0.054$	$0.374$	$0.101$	$0.378$	$0.454$

Appendix

Table 12 : Comparison of Out-of-Distribution (OOD) Mean Squared Error (MSE) in radians squared (

\text{rad}^{2}

) for our method and RoboPEPP. Results are shown per joint under various experimental conditions.

Panel of Figure 12 — Figure 12 : A step-by-step visualization of the reconstruction process for an occluded input. The final reconstruction from autoencoder-2, generated by passing its output through the decoder of autoencoder-1, successfully inpaints the occluded region.

Figure 13 : Evolution of MCC scores during training for AE2 using a 2-camera setup on the independent dataset. The plot compares performance across various learning rates. Training is performed with a batch size of 128, and sparsity loss weight of $\lambda=3$ .

Figure 14 : Evolution of MCC scores during training for AE2 using a 2-camera setup on the independent dataset. The plot compares performance across various batch sizes. Training is performed with a learning rate of $5e-5$ and sparsity loss weight of $\lambda=3$ .

Figure 15 : Evolution of MCC scores during training for AE2 using a 2-camera setup on the independent dataset. The plot compares performance across various combinations of reconstruction loss weights and sparsity loss weights. Training is performed with a batch size of 128 and a fixed learning rate of $5e-5$ .

Figure 16 : Ablation showing the advantage of having residual connections for better reconstruction loss and MCC score

	Joint 1		Joint 2		Joint 3		Joint 4		Joint 5		Joint 6
Calibration Samples	MCC	MSE	MCC	MSE	MCC	MSE	MCC	MSE	MCC	MSE	MCC	MSE
1000 samples	$0.874$	$0.083$	$0.979$	$0.015$	$0.634$	$0.217$	$0.950$	$0.035$	$0.679$	$0.198$	$0.884$	$0.080$
500 samples	$0.863$	$0.094$	$0.979$	$0.015$	$0.618$	$0.234$	$0.945$	$0.037$	$0.677$	$0.191$	$0.895$	$0.075$
100 samples	$0.881$	$0.079$	$0.975$	$0.019$	$0.612$	$0.227$	$0.942$	$0.039$	$0.661$	$0.199$	$0.879$	$0.084$

Appendix

Table 13 : MCC and MSE of our method across different Calibration settings. MSE is reported in radians squared.

	Joint 1		Joint 2		Joint 3		Joint 4		Joint 5		Joint 6
Lighting Condition	MCC	MSE	MCC	MSE	MCC	MSE	MCC	MSE	MCC	MSE	MCC	MSE
original	$0.874$	$0.083$	$0.979$	$0.015$	$0.634$	$0.217$	$0.950$	$0.035$	$0.679$	$0.198$	$0.884$	$0.080$
bright	$0.874$	$0.072$	$0.978$	$0.015$	$0.640$	$0.207$	$0.950$	$0.033$	$0.677$	$0.180$	$0.874$	$0.094$
dark	$0.811$	$0.107$	$0.940$	$0.046$	$0.520$	$0.211$	$0.860$	$0.093$	$0.199$	$0.303$	$0.526$	$0.265$

Appendix

Table 14 : MCC and MSE of our method across different lighting condition. MSE is reported in radians squared.

Panel of Figure 19 — Figure 19 : Visual comparison of the reconstruction quality at each stage of our pipeline for the single camera setup. (a) The original input image. (b) The reconstruction from the first autoencoder (AE1). (c) The final reconstruction from the second autoencoder (AE2)

Panel of Figure 20 — Figure 20 : Visual comparison of the reconstruction quality at each stage of our pipeline using the two camera independent model. (a) The original input image. (b) The reconstruction from the first autoencoder (AE1). (c) The final reconstruction from the second autoencoder (AE2)

Component	Layer-wise Details
Block Def.	ResBlockGN(f) :
	GroupNorm $\to$ ReLU $\to$ Conv(features=f, ks=3, pad=’SAME’)
	$\to$ GroupNorm $\to$ ReLU $\to$ Conv(features=f, ks=3, pad=’SAME’)
	$\to$ Add residual input
	(Note: ‘ks‘=kernel size, ‘s‘=stride, ‘pad‘=’SAME’)
Encoder	Input: Image $X\in\mathbb{R}^{128\times 128\times 1}$

Appendix

Table 15: Autoencoder 1 architecture ResNet-style with GroupNorm

Component	Layer-wise Details
Input Processing	The input shape depends on the camera setup:
	Single-Camera: Input $y\in\mathbb{R}^{8\times 8\times 1}$ is used directly.
	Two-Camera: Two feature maps $y$ are concatenated to form an input $\in\mathbb{R}^{8\times 8\times 2}$ .
Core Architecture	The following layers are applied to the processed input:
	Conv(features=32, ks=3), ReLU // Spatial dim: 8x8 $\to$ 6x6
	Conv(features=64, ks=3), ReLU // Spatial dim: 6x6 $\to$ 4x4

Appendix

Table 16: LDR Network Architectures for Single- and Two-Camera Setups.

Component	Layer-wise Details
Block Def.	ResBlockGN(f) :
	GroupNorm $\to$ ReLU $\to$ Conv(features=f, ks=3, pad=’SAME’)
	$\to$ GroupNorm $\to$ ReLU $\to$ Conv(features=f, ks=3, pad=’SAME’)
	$\to$ Add residual input
	(Note: ‘ks‘=kernel size, ‘s‘=stride, ‘pad‘=’SAME’)
Encoder	Input: $y\in\mathbb{R}^{8\times 8\times C_{in}}$ , where $C_{in}$ is 1 (single-cam) or 2 (two-cam).

Appendix

Table 17: Autoencoder 2 (AE2) Architecture.

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